Theorem cfinufil 24240

Description: An ultrafilter is free iff it contains the Fréchet filter cfinfil 24205 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
cfinufil	⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑋

Proof of Theorem cfinufil

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwi 4564	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
2		ufilb 24218	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 ∈ 𝐹 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
3	2	adantr 486	. . . . . . . . 9 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → (¬ 𝑥 ∈ 𝐹 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
4		ufilfil 24216	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
5	4	adantr 486	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → 𝐹 ∈ (Fil‘𝑋))
6		filfinnfr 24189	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑋 ∖ 𝑥) ∈ 𝐹 ∧ (𝑋 ∖ 𝑥) ∈ Fin) → ∩ 𝐹 ≠ ∅)
7	6	3exp 1137	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ((𝑋 ∖ 𝑥) ∈ Fin → ∩ 𝐹 ≠ ∅)))
8	7	com23 87	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ Fin → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ∩ 𝐹 ≠ ∅)))
9	5, 8	syl 18	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ Fin → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ∩ 𝐹 ≠ ∅)))
10	9	imp 412	. . . . . . . . 9 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ∩ 𝐹 ≠ ∅))
11	3, 10	sylbid 243	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → (¬ 𝑥 ∈ 𝐹 → ∩ 𝐹 ≠ ∅))
12	11	necon4bd 2976	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → (∩ 𝐹 = ∅ → 𝑥 ∈ 𝐹))
13	12	ex 418	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ Fin → (∩ 𝐹 = ∅ → 𝑥 ∈ 𝐹)))
14	13	com23 87	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (∩ 𝐹 = ∅ → ((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
15	1, 14	sylan2 605	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → (∩ 𝐹 = ∅ → ((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
16	15	ralrimdva 3163	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ → ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
17	4	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
18		uffixsn 24237	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → {𝑦} ∈ 𝐹)
19		filelss 24164	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ {𝑦} ∈ 𝐹) → {𝑦} ⊆ 𝑋)
20	17, 18, 19	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → {𝑦} ⊆ 𝑋)
21		dfss4 4215	. . . . . . . . . . 11 ⊢ ({𝑦} ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
22	20, 21	sylib 221	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
23		snfi 9064	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
24	22, 23	eqeltrdi 2869	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin)
25		difss 4083	. . . . . . . . . . 11 ⊢ (𝑋 ∖ {𝑦}) ⊆ 𝑋
26		filtop 24167	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
27		elpw2g 5295	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐹 → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
28	17, 26, 27	3syl 19	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
29	25, 28	mpbiri 261	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋)
30		difeq2 4068	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → (𝑋 ∖ 𝑥) = (𝑋 ∖ (𝑋 ∖ {𝑦})))
31	30	eleq1d 2846	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → ((𝑋 ∖ 𝑥) ∈ Fin ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin))
32		eleq1 2849	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → (𝑥 ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
33	31, 32	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → (((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) ↔ ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
34	33	rspcv 3573	. . . . . . . . . 10 ⊢ ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
35	29, 34	syl 18	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
36	24, 35	mpid 45	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝐹))
37		ufilb 24218	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑦} ⊆ 𝑋) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
38	20, 37	syldan 603	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
39	18	pm2.24d 152	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (¬ {𝑦} ∈ 𝐹 → ¬ 𝑦 ∈ ∩ 𝐹))
40	38, 39	sylbird 263	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝐹 → ¬ 𝑦 ∈ ∩ 𝐹))
41	36, 40	syld 48	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ¬ 𝑦 ∈ ∩ 𝐹))
42	41	impancom 457	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)) → (𝑦 ∈ ∩ 𝐹 → ¬ 𝑦 ∈ ∩ 𝐹))
43	42	pm2.01d 192	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)) → ¬ 𝑦 ∈ ∩ 𝐹)
44	43	eq0rdv 4365	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)) → ∩ 𝐹 = ∅)
45	44	ex 418	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ∩ 𝐹 = ∅))
46	16, 45	impbid 215	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
47		rabss 4018	. 2 ⊢ ({𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹 ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹))
48	46, 47	bitr4di 292	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  {crab 3413   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  ∩ cint 4907  ‘cfv 6537  Fincfn 8966  Filcfil 24157  UFilcufil 24211

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1o 8469  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fbas 21668  df-fg 21669  df-fil 24158  df-ufil 24213

This theorem is used by: (None)